Although different approaches have been implemented for social network simulation, we can classify them into two groups: analytical simulations using a continuous model and network-model based simulations. Analytical simulation tries to provide a model formulated into an integro-differentiable system. Unfortunately the complexity of such a system grows quickly for the analytical representation of multivariable and complex phenomenon of stochastic processes quickly become very difficult to manage in terms of computation and stability of the equations. The second approach, the network-model based simulation, is to see the problem from a different angle and to try to reach a solution step by step by step while affecting the behavior of various objects composing the model. It is not based on an analytical representation, but more on objects representation. This agent based simulation belongs to this category. This type of study is usually split into two parts: the first part deals with the structure and behavior of the population which we are working with; the second part is the intrinsic properties and characteristics of the disease which we are studying.
It is essential to note that in Sexual Transmitted Diseases (STD) simulations, the model of the sexual network is extremely important. The properties and rules that the HIV epidemic follows can be described into a simple model. Therefore we will emphasize on the previous sexual network studies in this section.
Sociologists and statisticians have been studying social networks for more than 50 years. Social Network Analysis bring and important insight on how to conceptualize and model social interactions. The sexual network is one kind of social network. In standard epidemiological models, random interactions among people are often assumed, especially if the medium of transmission is the air. In Sexual Transmitted Diseases, randomness does not stand anymore for sex does not happen by haphazard. Therefore a first approach of in STD simulation was to divide the population into subgroups defined according to various criteria. Those criteria can be age, ethnicity, social economical class, short time or long time partners, heterogenic degrees of sexual activity for instance. [3]
It is easier to deal with a population divided into subgroups, usually referred as core groups, and study the interaction among those groups. Three types of scenario emerge from this model. In the first one, called assortative interaction, most contacts take place inside subgroups; this means that sexually active people have sex with other sexually active people; whereas sexually low active people only have sex with other sexually low active people. In the second scenario, called disassortative interaction, most contacts take place in between the subgroups; this means that sexually active people have sex with sexually low active people. Theoretical studies of these models have demonstrated that assortative interaction generates a faster epidemic with a lower size. Disassortative interaction generates a larger size of epidemic, but the spreading is slower. In a mix scenario where interaction takes place both inside the subgroups and in between the subgroups, also called symmetric interaction, the result of speed and size of spreading of the epidemic is somewhere between the two previous models. Groups have different network patterns depending of the type of individuals belonging to this group. Figure 2.1 and 2.2 illustrate the different network shapes between sexually active and sexually low active people.
Figure 2.1: Sexually active people maintain several links with their partners. This structure is a favorable field to induce a fast spreading of the epidemic once the central node is infected.
Figure 2.2: Sexually low active people generate that kind of long structure.
They tend to include more nodes that the sexually active people network, which therefore explains why the size of the epidemic is larger.
Most of empirical studies indicate that assortative interaction is closer to reality. And exception arises from the study of the male homosexual population in Iceland which shows strong disassortative patterns. [4].
It is interesting to take a closer look at the results of empirical studies of sexual networks. A study based on clinical contact tracing and volunteered information associated with contact tracing in Manitoba, Canada, is one of the rare example of reconstructed sexual network. Surveyed subjects are who are confirmed to be positive to a STD (a total of 4544 individuals in this study) are asked to list their sexual partners.
Those partners are as many potentially infected subjects. These subjects are in turn, contacted, tested and asked to provide their own sexual partners, and so on until we can reconstruct part of the sexual network. Of course this kind of study relies on volunteered information and data are extremely difficult to obtain. The method is also arguable as the reconstruction of the sexual network relies only on the infected subjects and does not bring insight about the complete picture of the sexual network. Nevertheless, it has revealed that sexual networks are constituted of many relatively small clusters. The interesting point is to find out how those small clusters overlapped each other and foster the spreading of the epidemic. [5]
Sexual Networks are not static. Partnership evolves as time goes by, following different scales for different individuals. It is therefore necessary to consider the concurrent relationships on a network. [6] These studies have been developing new models to represent the nature of relations, involving fidelity among partners and transmission among two non-concurrent relationships that would happen relatively closely in time. Sexually low active populations tend to be more static whereas sexually active populations are much more dynamic. The influence of sexually active population in a network plays a major role in the spreading of sexual infections. Therefore the precision of the model of this subpopulation is of major importance.
A modern approach to study social networks is based on the Small World Properties.
In sexually transmitted diseases, there is contagion between two nodes if those nodes are
linked together. Most of the sexual networks have a very low average number of connections to other nodes but with a very big variance. In other words, most of the people in a sexual network have very few partners, but a small number of them have a lot of sexual partners. This architecture of network is closely related to Small World networks.
In a Small World network, most of the nodes have a close connectivity with their closest neighbors, therefore influencing the network on a local level, but only few nodes have a longer connection, linking randomly other parts of the network, therefore bringing two different regions closer to each other. (Figure 2.3 and 2.4) The interesting property of this kind of network is that the average distance between two random nodes is dramatically decreased due to the presence of these few nodes with longer links.
Figure 2.3: In a regular network, on node on one side of the network need a lot of intermediate nodes before reaching its opposite node on the other side of the ring. They stand at a far distance from each other.
Figure 2.4: In a Small World, a few nodes connect other parts of the network located far away from them. Therefore, two opposite nodes are now closer to each other thanks to the longer links that exist between two remote regions in the network.
It is possible to classify Small World networks into different sub-classes of networks.
These sub-classes are categorized according to the degree of distribution P(k) of a link to a node. Three classes of networks have been identified; single-scale, broad-scale and scale-free networks. The sexual networks belong to the category of scale-free networks.
[1][7][8]. In a scale-free network, most of the nodes have a small number of connections to other nodes, whereas a few nodes only actually have an important number of connections. It is possible to represent different categories of scale-free network just by changing the distribution for connectivity between nodes. The following graph (figure 2.5) shows the degree of distribution P(k) depending on the number of links k if a power law or a Poisson law is applied.
Figure 2.5: Comparison of the functional form of a Poisson distribution and a power-law distribution for connectivity, k, with logged axes. Studies suggest that the connectivity distribution for sexual contacts might follow a power law, thus indicating a scale-free network structure.
The sexual network following a scale-free pattern explains how an epidemic can be spread quickly on a large scale. The size and speed of the epidemic being determined by the law used to model the distribution of links. A high number of highly connected nodes will speed up the spreading of the epidemic on a large scale. That is the reason why the
distribution of connectivity plays a major role while attempting to build up a social network model.
Therefore the challenges in design and analysis of the HIV/AIDS epidemic relies in the sexual network study, given that data are extremely difficult to obtain.